The furnace is cycling strangely.

I had put on a new thermostat and then put the old one back, same issues
The furnace has a Honeywell S8600M ignighter

I shut everything off and after about an hour I turned tit back on

First the small exhust fan started
The main fire started and stayed on for about a minute before shutting off
Then the main blower fan came on and the fire started
The fire cycled on and off about every 15 to 20 secs
Then the part that bothers me is that the fire would come on and the main blower would stop
The fire would burn for 20 secs with only the small exhaust fan blowing
Then the flam would go out for 15 secs
The main blower would come on and blow for another 20 secs before the flame came back on and started the cycle all over again

Is this normal?

No

Its time to get it checked out by a service tech. From the symptoms you gave you could have a crack or hole in the heat exchanger putting you or your family in danger!!